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Symbolic dynamics originated as a method to study general dynamical systems; now its techniques and ideas have found significant applications in data storage and transmission, linear algebra, the motions of the planets and many other areas [citation needed]. The distinct feature in symbolic dynamics is that time is measured in discrete intervals.
One of Hedlund's early results was an important theorem about the ergodicity of geodesic flows. [7] He also made significant contributions to symbolic dynamics, whose origins as a field of modern mathematics can be traced to a 1944 paper of Hedlund, and to topological dynamics.
A symbolic flow or subshift is a closed T-invariant subset Y of X [3] and the associated language L Y is the set of finite subsequences of Y. [ 4 ] Now let A be an n × n adjacency matrix with entries in {0, 1}.
Pages in category "Symbolic dynamics" The following 8 pages are in this category, out of 8 total. This list may not reflect recent changes. ...
Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator.
Bowen: "Symbolic Dynamics for Hyperbolic Flows" in Proceedings of the International Congress of Mathematicians (Vancouver, 1974), pp. 299–302. Bowen: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. (Lecture Notes in Mathematics, no. 470: A. Dold and B. Eckmann, editors). Springer-Verlag (Heidelberg, 1975), 108 pp.
The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit . As the system evolves, points in this disk remain close to the given periodic orbit, tracing out orbits that eventually intersect the disk once again.
In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and symbolic dynamical systems are often considered synonyms. The most widely studied shift spaces are the subshifts of finite type and the sofic shifts.